Advertisements
Advertisements
Question
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
Options
increases on [0, a]
decreases on [0, a]
increases on [−a, 0]
decreases on [a, 2a]
Advertisements
Solution
Given: ϕ(x) = f(x) + f(2a − x)
Differentiating above equation with respect to x we get,
ϕ'(x) = f'(x) − f(2a − x) .....(1)
Since, f''(x) > 0, f'(x) is an increasing function.
Now,
when \[x \in \left[ 0, a \right]\]
\[f'\left( x \right) \leq f\left( 2a - x \right) . . . . . \left( 2 \right)\]
Considering equation (1) and (2) we get,
ϕ'(x) ≤ 0
⇒ ϕ'(x) is decreasing in [0, a]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the logarithmic function is strictly increasing on (0, ∞).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
The interval in which y = x2 e–x is increasing is ______.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function f(x) = xx decreases on the interval
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function f(x) = x9 + 3x7 + 64 is increasing on
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
The function `1/(1 + x^2)` is increasing in the interval ______
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
The function f(x) = x3 + 3x is increasing in interval ______.
Let \[x_0\] be in the domain of a real-valued function \[f\]. When is \[f\] increasing at \[x_0\]?
For \[f'(x)=12(x-3)(x+2)\], what are the critical points obtained from \[f'(x)=0\]?
